how do I work with LAB color space in the compositor?
You can't, not without adding a whole different reference space, and you don't actually want to. The explanation for this is very long winded, and I'm willing to explain, but the short version is as follows.

Proper mixing of colour relies on modeling our real-world of physics. That part includes using linearized systems that orbit around the spectral locus. Lab is strictly a perceptually uniform model based off of XYZ. As such, it is a nonlinear space, which makes it absolutely one of, if not the, worst possible choice for manipulation of images. No really, it is that bad[1].

So what you are actually requesting is a method to manipulate a dual axis model that works in conjunction with the default tristimulus (RGB) linearized reference model. At the current time, there is no OpenColorIO node, so you can't transform the data along the nodal pipeline.

Further, even if you could perform an OpenColorIO transformation along the pipeline, this would be an awful idea to take your reference RGB data to Lab for the reasons identified above. So is there a reasonable dual axis system that could work in conjunction with a linearized tristimulus reference space? In theory yes, you could change to xyY, which is a scaled version of XYZ. The problem with xyY is that the range of values that can be produced from the primary lights are largely imaginary, and as such, there are many other pitfalls that come along with that sort of a choice that have to be closely managed.

Perhaps sometime soon on the horizon, Blender will be fully agnostic and the colour pickers and such will be properly set up to enable a changing of viewing transforms on the UI elements. This would allow you to use whatever colour transformation you wish only on the view of the UI element, while leaving the actual reference values in their established state. That would permit using something like a Lab 3D LUT to view the picker, and give you the RGB values transformed for the reference space.

The key takeaway here is:


Lab is an awful choice for manipulation of images, and a questionable choice for colour based selection.
A clear division of reference / working space and UI / View space alleviates any artistic reasoning for (1) above. That is, as long as your reference remains strictly scene linearized, and the view / UI permits one-way only transformation of those values, the correctness of the image manipulation pipeline is preserved.


[1] The great granddaddy of misinformation was a tome written for a commercial image manipulation program that has since become infamous among imagers. It has some of the worst advice, some of the biggest misinformation, and some of the most dire concepts delivered to imagers in perhaps the past twenty years. Lab is a fool's errand.